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QUIVALENCE  AND  REDUCTION   OF   PAIRS 
OF  HERMITIAN  FORMS 


A  DISSERTATION 

BMITTBD    TO    THK    FACULTY    OF    THE    OGDEN    GRADUATE    SCHOOL  OF  SCIENCE  IN 
CANDIDACY  FOR  THE  DEGREE  OP  DOCTOR  OF  PHILOSOPHY 
DEPARTMENT  OF  MATHEMATICS 


HV 

MAYME  IRWIN  LOGSDON 


n^ 


Private  Edition,  Distributed  By 

The  University  of  Chicago  Libraries 

Chicago,  Illinois 


Repritited  from  , 

Ambrican  Journal  of  Mathematic^s 
Vol.  XLIV,  No.  4,  October,  1<)22 


Ztbe  1Ilntver0tt^  ot  Cbi'cadii :..  ii* '0 1^ '{.:  i  v. 


EQUIVALENCE  AND  REDUCTION   OF   PAIRS 
OF  HERMITIAN  FORMS 


A  DISSERTATION    ' 

SUBMITTED    TO    THE    FACULTY    OF    THE    OGDEN    GRADUATE    SCHOOL  OF  SCIENCE  IN 

CANDIDACY  FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 

DEPARTMENT  OF  MATHEMATICS 


BY 

MAYME  IRWIN  LOGSDON 

M 


Private  Edition,  Distributed  By 

The  University  of  Chicago  Libraries 

Chicago,  Illinois 


Reprinted  from 

American  Journal  of  Mathematics 

Vol.  XLIV,  No.  4.  October,  1922 


U 


eXCHAlvuv. 


Reprinted  from  The  American  Journal  op  Mathematics,  Vol.  XlAV,  Wo.  4,'6ct6b'erl  'i9^ 


EQUIVALENCE    AND    REDUCTION    OF    PAIRS    OF    HERMITIAN 

FORMS.* 

By  Mayme  Irwin  Logsdon. 

Introduction  and  Summary. — Of  fundamental  importance  in  the  theory 
of  matrices  and  forms  has  been  the  use  by  Weierstrass,  Kronecker,  and 
Frobenius  of  the  theory  of  elementary  divisors  in  the  study  of  equivalence 
and  reduction  of  pairs  of  bilinear  and  pairs  of  quadratic  forms. 

In  this  paper  a  generalization  is  made  in  that  the  basal  theorems  of  the 
theory  are  extended  to  any  hermitian  X-matrix,  i.e.,  a  matrix  whose  elements 
are  polynomials  of  degree  w  in  X  with  coefficients  in  the  field  of  complex 
numbers  and  are  such  that  the  conjugate  of  the  element  in  the  ith  row  and 
jth  column  is  equal  to  the  element  in  the  jth  row  and  ^th  column  for 
i,j=  1,  2,  "■,n. 

Inasmuch  as  a  linear  substitution  with  matrix  P  on  the  variables  of  an 
hermitian  form  with  matrix  a  gives  a  form  with  matrix  b  =  P'aP  where  P 
means  the  matrix  formed  from  P  by  taking  the  conjugate  imaginary  of 
each  element  and  P'  means  the  transposed  matrix  P,  the  extension  of  the 
general  theory  to  the  hermitian  X-matrix  is  justified  by  the  proof  in  Part  I  of 

Theorem  II .  If  h  =  p'aq  where  p  and  q  are  non-singular  and  inde- 
pendent of  X,  and  where  a  and  b  are  hermitian  \-matrices,  then  there  exists  a 
matrix  P  such  that  b  =  P'aP. 

The  special  application  is  made  to  hermitian  X-matrices  whose  elements 
are  linear  in  X.     Such  will  be  the  matrix  of  the  pencil  of  forms 

n       n 

\A  —  B  =  ^Y^  Q<aij  —  bij)xiXj. 
1=1  y=i 

The  coincidence  of  the  elementary  divisors  is  found  to  be  a  necessary 
and  sufficient  condition  for  the  equivalence  of  two  pairs  of  hermitian 
matrices  free  of  X  and  for  the  equivalence  of  two  pairs  of  hermitian  forms. 

In  Part  II,  the  Weierstrass  reduction  is  shown  to  hold  in  case  one  of 
the  forms  is  definite,  a  condition  which  insures  reality  of  all  the  elementary 
divisors;  in  fact  the  Weierstrass  method  can  be  used  for  finding  the  con- 
tribution to  the  canonical  form  of  any  real  elementary  divisor.  In  the 
case  however  of  conjugate  complex  elementary  divisors,  (X  —  d)'  and 
(X  —  a)*,  it  was  found  necessary  and  possible  to  regularize  the  X-matrix 
with  respect  to  the  two  conjugate  imaginary  linear  factors  simultaneously 

*  Presented  to  the  Society  at  Chicago.  March  25,  1921. 

247 


52nr)i):i 


248i  i  **  •** .  *.**.•  •     Lqqsd^n:   Pairs  of  Hermitian  Forms. 

and  also  to  expand  the  terms  representing  the  adjoint  form  with  respect  to 
these  two  factors  simultaneously. 

In  the  actual  work  of  reduction  use  was  made  of  an  algebraic  simplifica- 
tion suggested  and  described  by  Dickson*  in  "  Pairs  of  Bilinear  or  Quadratic 
Forms."  The  importance  of  this  simplification  is  that  in  the  case  of  bilinear 
forms  the  reduction  may  be  accomplished  rationally  while  in  the  case  of 
quadratic  or  hermitian  forms  the  computations  are  simplified  if  the  con- 
stants c^  which  appear  are  held  until  the  last  step  of  the  reduction  before 
being  absorbed  in  the  variables. 

The  canonical  form  obtained  f  is  given  in  Part  I,  Theorem  V. 

Part  I — Equivalence. 
We  seek  the  conditions  for  equivalence  of  two  pairs  of  hermitian  forms : 

n       n  n       n 

A  =    /  ,    /  J  dijXiXj,  A      =    /  J    /  J  OiiiXiXj, 

i=i  }=i  t=i  y=i 

n        n  n        n 

5  =  S  £  bijXiXj,        5*  =  X)  Z)  ^IjXiXj, 
1=1  y=i  1=1  i=i 

with  the  respective  matrices  a,  a*,  b,  b*  of  which  b  and  b*  are  non-singular, 
and  where  a,y  =  aji,  etc.,  for  i,j=  1,  •  •  •,  n. 

If  a  linear  transformation  with  non-singular  matrix,  c,  whose  elements 
are  independent  of  X, 

n 

Xi  =  J^Cijyj        {i=  1,  •■•,n), 
i=i 

with  the  induced  transformation  on  the  conjugate  variables, 

n 

Xi  =  S  CijVj         (i  =  1,  ■••,n), 
i=i 

be  applied  to  A  and  B,  the  transformed  hermitian  forms  will  have  the 
respective  matrices  c'ac  and  c'bc.     If,  now,  these  are  to  be  the  forms  A* 
and  B*,  we  must  have 
(1)  a*  =  c'ac,        b*  =  c'bc. 

We  shall  show  that  a  necessary  and  sufficient  condition  that  equations  (1) 
be  satisfied  is  that  the  two  hermitian  X-matrices,  m  =  a  —  \b  and  m* 
=  a*  —  Xb*;  have  the  same  elementary  divisors.  We  state  the  problem 
thus: 


.     *  Transactions  A.  M.  Society,  v.  10,  1909,  p.  350. 

t  In  the  Proceedings  of  the  London  Mathematical  Society,  v.  32,  1900,  pp.  321 , 

Bromwich  obtains  such  a  reduction  by  a  special  device,  stating  that  "apparently  this 
method  (the  Frobenius-Weierstrass  method)  cannot  be  extended  so  as  to  cover  the  analogous 
theory  for  conjugate  imaginary  substitutions,  which  would  be  applied  to  a  pair  of  Hermite's 
forms." 


LoGSDON :   Pairs  of  Hermitian-'Fhrj^^     ; . . . .     ••....  24$ 

Given  any  two  hermitian  \-matrices,  m  and  m*,  toith  elements  polynomials 
in  \,  to  find  necessary  and  sufficient  conditions  for  the  existence  of  a  non- 
singular  matrix  c  untk  elements  independent  of  X,  such  that  m*  =  c'mc. 

In  proof  we  must  show  that  if  two  hermitian  X-matrices  are  equivalent, 
the  corresponding  hermitian  forms  may  be  obtained,  the  one  from  the 
other,  by  a  non-singular  transformation  on  the  variables.  The  first  step 
of  the  proof  will  consist  in  establishing 

Theorem  I.  If  two  hermitian  \-matrices,  m=a—\b  and  m*=a*— X&*, 
are  equivalent,*  there  exist  two  non-singular  matrices,  t  and  q,  whose  elements 
are  independent  of  X,  such  that 

(2)  m*  =  tmq. 

Proof:  By  the  equivalence  of  m  and  m*  there  exist  non-singular  X-matrices 
(q  and  Qo  with  determinants  free  of  X,  such  that 

(3)  m*  =  tomqo. 

Now  divide  ^o  by  m*  and  (qo)~^  by  m  f  in  such  a  way  as  to  get  matrices 
ti,  t,  Si,  s  which  satisfy  the  relations 

(4)  to  =  m*ti  +  t,         (qo)-^  =  Sim  +  s, 

t  and  s  being  matrices  whose  elements  do  not  involve  X.  From  (3)  we  get 
torn  =  m*q^^.     Substituting  here  from  (4)  we  have  " 

(5)  m*{ti  —  Si)m  =  m*s  —  tm. 

Now  the  right  member  of  (5)  is  a  X-matrix  of  at  most  the  first  degree,  while 
for  ti  —  Si  y>^  0  the  left  member  would  be  of  at  least  the  second  degree. 
Hence  ti  =  Si  and 

(6)  m*s  =  tm. 

Whence  if  we  knew  that  s  (and  likewise  t  from  (6))  were  non-singular, 
we  could  write 

(7)  m*  =  tms~^ 

and  the  theorem  would  be  proved. 

We  proceed  to  show  that  s  is  non-singular.     Substitute  in  the  identity 

I  =  QoQ6~^  for  qo~^  from  (42)  and  get 

(8)  /  =  qoSiW.  H-  qos. 
Now  divide  qo  by  m*  in  such  a  way  as  to  get 

(9)  qo  =  qm*  +  q, 

*  Two  X-matrices,  m  and  m*,  are  called  equivalent  (,B6cher,  Introduction  to  Higher 
Algebra,  p.  274)  if  there  exist  X-matrices  to  and  ^o  each  having  as  determinant  a  number  not 
zero  independent  of  X  such  that  m*  —  Umqo. 

t  B6cher,  p.  278. 


250  ;.'.•'     .••; :     L(JP^&n:   Pairs  of  Hermitian  Forms. 

where  g  is  a  matrix  with  elements  free  of  X.     Substituting  this  value  in  (8) 
we  have 

/  =  q^Sim  +  qim*s  +  qs 

which,  by  use  of  (6),  may  be  written 

(10)  I  —  qs=  (qoSi  +  qit)m. 

Since  the  left  member  does  not  contain  X  we  must  have  qoSi  +  qit  identically 
zero,  and  therefore 

(11)  /  =  qs. 

s  is  then  non-singular,  and  we  may  write  (6)  in  the  form  * 

m*  =  tmq. 

Since  p  =  t'  is  evidently  a  X-matrix  whose  determinant  equals  the  con- 
jugate of  the  determinant  t,  we  may  express  the  definition  of  equivalence 
in  the  following  form  which  is  more  convenient  for  hermitian  forms: 

Two  hermitian  X-matrices,  m  and  m*,  are  equivalent  if  there  exist  non- 
singular  matrices  p  and  q  with  determinants  free  of  X,  such  that  m*  =  p'mq. 

The  second  step  in  solution  of  the  original  problem  makes  possible  the 
extension  of  the  theory  of  equivalence  to  the  corresponding  forms.  It 
consists  in  proving 

Theorem  II.  If  b  =  p'aq,  where  p  and  q  are  non-singular  and  inde- 
pendent of  X,  and  where  a  and  b  are  hermitian  \-matrices,  then  there  exists  a 
matrix  P  such  that 

b  =  P'aP 

and  such  that  P  depends  not  on  a  or  b  but  solely  on  p  and  q. 
We  have  by  hypothesis 

(1)  b  =  p'aq, 
whence,  since  a  and  b  are  hermitian, 

b  =  q'ap. 
Equating  these  two  values  of  b  we  get 

(2)  q'ap  =  p'aq, 

from  which 

(3)  {q'r'p'a  =  apq-\ 

If  now  we  set  U  =  {q')~^p',  then  U'  will  be  pq~^  and  (3)  becomes 

(4)  Ua  =  aU. 

*  This  theorem  holds  if  m  and  m*  have  elements  of  degree  p  in  X. 


LoGSDON :   Pairs  of  Hermitian'Forjiis^ 


.251 


t/2a  =  aU'''; 


From  this  we  get  at  once 

(5) 

and,  in  general, 

(6) 

From  a  =  a  and  (4),  (5),  (6)  by  using  any  set  of  arithmetical  multipliers 
we  get 

(7)  x{U)a  =  ax(u'), 

where  x(^  is  any  polynomial  in  U.    Thus 

(70  a  =  lx(U)T'axm. 

Now  we  choose  the  polynomial  x(^  =  ^  so  that  V^  =  U  and  so  that  V  is 
non-singular.*    We  have  then  from  (7') 


(8) 

Substituting  (8)  in  (1)  we  get 

(9) 


a  =  y-^aV. 


b  =  p'V-^aV'q. 

Now  set  P  =  V'q,  then  P'  =  q'V  and  h  =  P'aP,  as  desired.     For,  from 
the  definitions  of  U  and  V  we  have 

U=V''=  {q')-9 

from  which  we  easily  obtain  q'V  =  p' V~'^. 

These  theorems  permit  the  use  of  the  general  theory  of  X-matrices. 
We  state  the  theorems  and  definitions  which  are  needed  in  the  sequel: 

I.  If  a  and  h  are  equivalent  hermitian  X-matrices  of  rank  r,  and  if 
Z),(X)  is  the  greatest  common  divisor  of  the  i-rowed  determinants  {i  ^  r)  of 
a,  then  it  is  also  the  greatest  common  divisor  of  the  i-rowed  determinants 
of  b. 

II.  Every  hermitian  X-matrix  of  order  n  and  rank  r  can  be  reduced  by 
elementary  transformations  f  to  the  normal  form 


Ei(X)        0 
0        E2{\) 


0 
0 

^r(X) 

0 


where  the  coefficient  of  the  highest  power  of  X  in  each  of  the  polynomials 
EiQC)  is  unity,  and  E,(X)  is  a  factor  of  Ei+i(K)  for  i  =  1,  2,  •  •  •,  r  —  1. 
III.  The  greatest  common  divisor  of  the  i-rowed  determinants  of  an 

*  F  is  in  fact  a  polynomial  in  U  of  degree  less  than  n.     See  B6cher,  I.e.,  p.  299. 
t  Elementary  transformations  as  defined  in  B6cher,  I.e.,  p.  262. 


253..-..'     .";.«     L(>pSDON:   Pairs  of  Hermitian  Forms. 

hermitian  X -matrix  of  rank  r,  when  i  <  r,  is 

Di(K)  =  EME2(K)  •  •  •  Ei(K), 

where  the  E's  are  the  polynomials  of  the  last  theorem. 

IV.  A  necessary  and  sufficient  condition  that  two  hermitian  X-matrices 
of  order  n  be  equivalent  is  that  they  have  the  same  rank  r,  and  that  for 
every  value  of  /  from  1  to  r  inclusive,  the  i-rowed  determinants  of  one 
matrix  have  the  same  greatest  common  divisor  as  the  i-rowed  determinants 
of  the  other. 

From  the  definition  of  the  D's  in  III,  we  see  that 

(i=  1,2,  ...,r),         (Z)o(X))=  1). 

Hence  since  the  D's  with  the  rank  form  a  complete  system  of  invariants, 
since  the  D's  completely  determine  the  E's  as  w^ell  as  the  elementary 
divisors,  and  since  conversely  the  D's  are  completely  determined  by  the  E's 
or  by  the  elementary  divisors,  we  may  state  thus  the 

Fundamental  Theorem:  A  necessary  and  sufficient  condition  that  two 
hermitian  \-matrices  be  equivalent  is  that  they  have  the  same  rank  and  that  the 
elementary  divisors  of  one  be  identical  respectively  with  the  elementary  divisors 
of  the  other. 

Definition:  Two  pairs  of  hermitian  matrices  a,  b  and  a*,  b*  with  elements 
free  of  X  will  be  called  equivalent  if  there  exist  two  non-singular  matrices 
p  and  q,  also  with  elements  not  involving  X,  such  that 

(1)  a*  =  Jag,        b*  =  p'bq. 

From  this  definition.  Theorem  I  and  the  fundamental  theorem  we  have 

Theorem  III.     7/  a,  b  and  a*,  b*  are  two  pairs  of  hermitian  matrices 

independent  of  X,  and  if  b  and  b*  are  non-singular,  a  necessary  and  sufficient 

condition  that  these  two  pairs  of  matrices  be  equivalent  is  that  the  two  \-matrices 

m  =  a  —  \b,        m*  =  a*  —  Xb*, 

have  the  same  elementary  divisors. 

Referring  now  to  Theorem  II,  equivalence  conditions  for  two  pairs  of 
matrices  may  be  stated  as  follows : 

Theorem  IV.  If  a,  b,  a*,  b*  are  hermitian  matrices  independent  of  X, 
and  if  b  and  b*  are  non-singular,  a  necessary  and  sufficient  condition  that  a 
non-singular  matrix  P  exist  such  that 

a*  =  P'aP,        b*  =  P'bP, 

is  that  the  matrices  a  —  'Kb  and  a*  —  X6*  have  the  same  elementary  divisors. 


Logsdon:   Pairs  of  Hermitfan-'Fotvii.:'  *..:  ••.*.  •*.•  •/.\253 

If  in  particular  b*  =  b  =  I,  where  /  is  the  unit  matrix,  we  have 

/=  PP 

which  defines  an  orthogonal  hermitian  matrix. 

Corollary:  If  the  characteristic  matrices  of  a  and  a*  have  the  same 
elementary  divisors  there  will  exist  an  orthogonal  matrix  P  such  that 

a*  =  P'aP        or        a*  =  P-^aP, 

i.e.,  a*  is  the  transform  of  a  by  the  orthogonal  matrix  P, 

We  have  thus  obtained  the  desired  conditions  for  equivalence  of  two 

pairs  of  hermitian  forms  as  stated  in  the  first  paragraph  of  Part  I. 

If  the  matrix  of  the  form  B  is  the  unit  matrix,  referring  to  the  last 

corollary  we  see  that  a  transformation  on  the  variables  with  orthogonal 

matrix  P  will  transform  the  form  B  into  B*  also  with  unit  matrix.     The 

X-matrices 

a  -  \I,        a*  -  X/ 

are  now  the  characteristic  matrices  of  the  forms  A  and  A*,  and  as  before 
a  necessary  and  sufficient  condition  for  the  equivalence  of  the  forms  under 
orthogonal  transformation  is  the  coincidence  of  the  elementary  divisors  of 
the  characteristic  matrices  of  the  forms. 

If  5  is  a  non-singular  definite  form,  a  preliminary  transformation  will 
transform  it  to  the  sum  of  hermitian  squares,  X!i^»^t  with  unit  matrix,  and 
since  the  roots  of  the  determinant  |  a  —  X6 1  =0  are  now  the  roots  of  the 
characteristic  equation  of  a  which  are  known  to  be  always  real,t  we  have 
the 

Corollary:  The  elementary  divisors  of  the  pencil  of  hermitian  forms 
A  —  \B,  where  B  is  non-singular  definite,  are  all  real  and  of  the  first  degree.  | 

As  in  the  quadratic  case  we  have  a  further 

Corollary:*  If  A  and  B  are  hermitian  forms  and  B  is  non-singular,  a 
necessary  and  sufficient  condition  that  it  be  possible  to  reduce  A  and  B 
simultaneously  by  a  non-singular  transformation  to  forms  into  which  only 
square  terms  (hermitian  squares)  enter  is  that  all  the  elementary  divisors 
of  the  pair  of  forms  be  of  the  first  degree. 

Finally,  if  both  matrices  are  singular  but  at  least  one  matrix  of  the 
pencil  Xifl  +  ^2b  is  non-singular,  we  may  proceed  as  in  the  quadratic  case  § 
and  obtain  the  desired  canonical  form. 

The  canonical  form. 

*  See  B6cher,  I.e.,  p.  30.5,  for  the  corresponding  theorem  for  quadratic  forms. 
t  G.  Kowalewski-Einfuhrung  in  die  Determinanten  Theoric,  p.  130. 
X  The  proof  in  B6cher,  I.e.,  p.  170,  for  quadratic  forms  is  applicable  here. 
§  Muth,  I.e.,  p.  87. 


254;/. :  '.•'.  ;.••{..*  ••traQSTBONy  Pairs  of  Hermitian  Forms. 

Theorem  V.  If  Ci,  Cg,  •••,  c/  are  any  real  constants  including  zero, 
equal  or  unequal,  if  ag,  an,  •  •  • ,  ttr  are  any  complex  numbers,  equal  or  unequal, 
and  if  e\,  e^,  •  •  • ,  -er  are  positive  integers  such  that  ^1  +  ^2+  •  •  •  +  ^/  +  2^^ 
+  •  •  •  +  2er  =  n,  there  exist  pairs  of  hermitian  forms  in  n  variables,  the 
first  form  being  non-singular,  which  have  the  elementary  divisors 

(X  -  ci)«s  (X  -  C2y\  '■-,  (X  -  c/)%  (X  -  a,)^  (x  -  a,y%  •  •  •, 

(X  -  drY^  (X  -  arY-. 
In  proof  after  setting  ^1  +  ^2+  •  •  •  +  e/  =  e  we  exhibit  the  forms 

ei    ei  +  ej    

■4  =    ^  XiXei—i+1  -\-      2^     XiX2ei+e2—i+l  +    '  '  ' 
1=1  i=ei  +  l 

e  «+2e^   

"h         2-^       XiX2e-i+l  +      2^    XjX23+2eg-j+l 
i=e—ej+l  ^-e+1 

n  

+    •  •  •   +  Z^         XjX2n—2s^-j+l- 

ei        ei+e2        

B   =    zL  ClXiXei-i+l  +  22     (^2^i^2ei+e^i+l  +    '  *  ' 

i=l  i=ei+l 

e  ei— 1  ei+€2— 1  

"1"         Z^       CfXiXze—i+l  +   Z-J  -^i^ei-i  +       2_/      -^i-^aei+ej-i  4"    *  "  ' 
e+e,,         e+2eg  

~^      Z^    f^gX jX^e+^e^—j+l  +         2-j       ^g-^j-^2e+2e^—j+l  +    "  *  * 
J=e+l  J=e+eg+l 

n—e^  n  

+          2^         arXjX2n-2e^-j+l  ~\~         ^        dfX jX2n—2e—j+\ 
j=n-2e^+\                                                  i=»-e^+l 
e+2€g-l  »— 1  

+       ZL      XjX2e+2e^-j  +    *  *  *   +  ZL         ^j^2n-2e^-]- 

J=e+1  j=n-2er+l 

Part  II — Reduction. 

In  the  attempt  to  apply  the  naethods  of  Weierstrass  to  reducing  a  pair 
of  hermitian  forms: 

n        n  n       n 

-4  =  X)  Zl  ttijXiXj        and         ^  =  Z)  S  bijXiXj, 
i=i  j=i  1=1  ^=1 

where  da  =  aji,  ba  —  bju  \aij\  9^  0,  no  trouble  arises  in  finding  the  con- 
tribution to  the  canonical  form  due  to  any  real  linear  factor  of  the  X-matrix, 
Xa  —  h,  though  proof  is  needed  that  certain  theorems  are  actually  extensible 
to  this  type  of  matrix.  In  dealing  with  complex  linear  factors  however  an 
essential  modification  is  required.  We  shall  first  indicate  the  main  steps 
in  the  process  of  reduction,  then  study  in  detail  the  separate  cases  where 
difference  of  treatment  is  required.  Wherever  the  Weierstrass  treatment 
as  given  in  Muth's  "Theorie  und  Anwendung  der  Elementartheiler  "  is  valid 
without  separation  of  the  two  cases,  the  details  of  the  work  will  be  omitted 


Logsdon:  Pairs  of  HermiUmcr/jrJiiii'  * .:  ••.;  •*.•  *  .*,*  255 

and  reference  to  Muth  given.  The  following  notations  and  definitions 
will  be  used : 

S  =  determinant  of  the  form  C  =  \A  —  B. 

l^  =  exponent  of  the  linear  factor  (X  —  r)  in  D^(K),  i.e.,  in  the  greatest 
common  divisor  of  all  the  K-rowed  minor  determinants  of  S.  There  will 
be  at  least  one  K-rowed  minor  determinant  of  S  which  contains  (X  —  r) 
exactly  l^  times  and  is  then  defined  to  be  regular  with  respect  to  this  factor. 
We  have 

and  also 

where  (X  —  r)'*  is  an  elementary  divisor. 

Sij  =  cof actor  of  the  element  Xajy  —  bij  in  S. 

S(«)  =  principal  minor  determinant  with  n  —  k  rows  obtained  from  S 
by  deleting  the  first  k  rows  and  the  first  k  columns.  We  note  S^*^  =  S^J-''~^\ 
For  K  =  0,  we  define  S(°>  =  S,  and  for  k  =  n,  S^">  =  1. 

Sp  =  the  principal  p-rowed  minor  determinant  in  the  upper  left-hand 
corner  of  the  matrix. 

Sp;  ik  =  (p  +  l)-rowed  minor  determinant  obtained  by  bordering  S^  by 
the  ith.  row  and  the  Arth  column  of  the  original  matrix. 

The  main  steps  in  the  reduction  are: 

(1)  Any  hermitian  X-matrix  may  by  elementary  transformations  *  be 
regularized  with  respect  to 

(1)  any  real  linear  factor,  (X  —  c). 

(2)  any  pair  of  conjugate  imaginary  linear  factors, 

(X  —  a),     (X  —  a). 
*  Elementary  transformations  of  an  hermitian  X-matrix  are  defined  as  follows: 

1st.     Interchanging  two  columns  and  the  same  two  rows. 

2d.  Multiplying  the  elements  of  a  column  by  a  number  m  independent  of  X  and  then 
multiplying  the  elements  of  the  corresponding  row  by  the  conjugate  of  m. 

3d.  Adding  to  the  elements  of  the  Ath  column  the  products  of  the  corresponding 
elements  of  the  jth  column  (J  9-^  k)  hy  a  polynomial,  ^(X),  then  adding  to 
the  elements  of  the  A;th  row  the  products  of  Jiie  corresponding  elements  of  the 
jth  row  each  multiplied  by  the  conjugate,  f  (X),  of  the  polynomial  ^(X). 

The  elementary  transformations  of  the  variables  of  the  corresponding  hermitian  form  which 
effect  on  the  matrix  of  the  form  the  above  defined  transformations  are: 


.^ 


1st.    Xi  =  Pi     (i  =  1,  •  •  •,  n;  i  j-^j,  i  7^  k) 

Xi  =  Vk 

Xk  =  Vi. 
2d.     Xi  =  yi     (i  =  1,  •'•,n;  i  9^  j) 

Xi  =  myi. 
3d.     Xi  =  yi     (i  =  1,  •  •  •,  n;  i  9^  j) 

X,  =  Vi  +  H^)yk. 


256 'V'  J  ';/]  '/\ '.,',  '''S^OGsi^'r' Pairs  of  Hermitian  Forms. 

Definition:  A  matrix  S  is  regular  with  respect  to  a  real  linear  factor  or 
with  respect  to  a  pair  of  conjugate  imaginary  linear  factors  if  each  of  the 
principal  minor  determinants  obtained  from  S  by  deleting  the  first  k  rows 
and  columns,  {k  =  1,  2,  •  •  •,  n  —  1),  is  so. 

(2)  The  adjoint  form  may  be  written  as  a  sum  of  terms  in  which  every 
factor  in  a  denominator  is  regular  with  respect  to  a  given  linear  factor 
(or  pair  of  factors) ;  viz., 

(1)  2-.  g  ^J^i        gg,  i-  g,g.  +  •  •  •  -h  ^(„-i)^(„)  • 

(3)  1st.  Each  term  on  the  right  of  (1)  may  be  expanded  with  respect 
to  a  real  linear  factor,  (X  —  c),  the  total  coefficients  of  1/(X  —  c)^,  1/(X  —  c) 
secured;  finally,  the  total  coefficients  of  1/X^,  1/X  secured.  This  will  give 
the  contribution  of  this  particular  linear  factor,  X  —  c,  to  the  canonical  form. 

2d.  Each  term  on  the  right  of  (1)  may  be  expanded  with  respect 
to  X  —  a,  X  —  a  simultaneously  and  the  total  coefficients  of  1/X^,  1/X 
secured. 

(4)  The  adjoint  form  may  be  expanded  by  determinantal  methods  in 
descending  powers  of  X  and  the  coefficients  of  1/X^,  1/X  obtained.  These 
prove  to  be  B  and  A  respectively. 

(5)  A  comparison  of  the  results  of  (3)  and  (4)  with  Theorem  V  of  Part  I 
gives  the  desired  expression  for  A  and  B. 

Proofs: 

(7i).  Any  hermitian  X-matrix,  S,  may  by  elementary  transformations 
be  regularized  with  respect  to  a  real  linear  factor,  X  —  c. 

In  proof  we  must  show 

(a)  Every  regular  p-rowed  minor  determinant  (p  >  1)  contains  at  least 
one  regular  (p  —  1) -rowed  minor  determinant  as  first  minor. 

(6)  Every  regular  (p  —  l)-rowed  minor  determinant  (p  >  2)  is  con- 
tained in  at  least  one  regular  p-rowed  minor  determinant  as  first  minor. 

(c)  If  a  (p  —  l)-rowed  principal  minor,  Sp_i,  is  regular,  but  no  p-rowed 
principal  minor,  *Sp_i;  <,  u  containing  it  is  regular,  there  is  an  elementary 
transformation  of  the  variables  which  without  disturbing  the  regularity  of 
any  Sa:  (^  =  1,  •  •  • ,  p  —  1)  will  so  transform  the  matrix  S  that  there  will 
be  a  regular  p-rowed  principal  minor  containing  Sp..i  as  a  first  minor. 

For  proof  of  (a)  and  (6)  see  Muth,  I.e.,  pp.  7-11. 

Proof  of  (c):  The  existence  of  a  regular  p-rowed  minor,  iSp_i;  j^ ,,,  con- 
taining Sp-i,  is  guaranteed  by  (6)  above.  Now  apply  to  the  variables  of 
the  form  a  preliminary  transformation  which  will  interchange  the  pth  and 
jth  rows  and  columns  and  the  (p  +  l)st  and  kih  rows  and  columns.  We 
now  have  Sp_i.  p_p+i  for  our  regular  p-rowed  minor.     Now  apply  the  trans- 


Logsdon:   Pairs  of  Hermitinjt  Ponnif'  *•.•  l'\  :'•:  :/•'•,  257 


formation 


Tm:  Xi  =  yi        {i  =  1,  •  •  •,  n;  i  =}=  p  +  1), 
Xp+i  =  2/p+i  -  TTiy^. 

The  effect  on  the  matrix  is  to  subtract  the  products  by  m  of  the  elements 
of  the  (p  +  l)st  column  from  the  elements  of  the  pth  column  and  then  the 
products  by  m  of  the  elements  of  the  (p  +  l)st  row  from  the  elements  of 
the  pth  row.  Obviously  Sk  is  not  changed  {k  =  1,  •••,  p—  1),  but  the 
principal  minor,  call  it  S'p,  of  order  p  and  containing  Sp_i,  is  now  regular. 
For  we  have 

Now  Sp  and  Sp_i;p4.i,  p+i  each  by  hypothesis  contains  X  —  c  more  than  /p 
times;  iSp_i;p, p+i  contains  X  —  c  exactly  Zp  times;  Sp_i.p+i^p  is  the  con- 
jugate of  Sp_i.pp+i  and  therefore  contains  the  real  linear  factor  X  —  c 
exactly  l^  times.  It  remains  then  to  show  that  the  sum  R  =  wSp_i.p  p+i 
+  wSp_i.p  p+i  does  not  have  a  higher  power  of  X  —  c  as  factor  than  each 
part,  for  every  m.  Divide  /?  by  (X  -  c)'".  i?  =  (X  -  c)^''[mf(X)  +  7n/(X)], 
and,  since  /(c)  is  not  zero,  if  we  set  m  =  l/fic)  we  have  mf{c)  +  mf(c)  not 
zero.     Thus  S^  contains  X  —  c  exactly  Ip  times  and  is  regular,  as  stated. 

(72)  Any  hermitian  X-matrix,  S,  may  by  elementary  transformations 
be  regularized  with  respect  to  an  imaginary  linear  factor.  When  this  is 
done  the  matrix  will  be  also  regular  with  respect  to  the  conjugate  imaginary 
linear  factor. 

As  before,  (a)  and  (6)  may  be  assumed  for  the  factor  X  —  a  from  the 
proof  for  bilinear  forms.  To  prove  (c)  we  apply  the  same  transformation, 
Tm,  and  get  as  before  the  right  member  of  equation  (3).  Remembering 
that  the  first  and  last  terms  are  principal  minors  and  hence  have  real 
coefficients,  and  that  neither  is  regular  with  respect  to  (X  —  a) (X  —  a),  we 
may  factor  the  right  member  of  (3)  thus: 

[(X  -  a)(X  -  a)]'''[(X  -  a)(X  -  a)J/i(X)  -  m(X  -  a^MX) 

-  m  (X  -  a)f2(X)  +  mml(\  -  a)(X  -  a)JUOC), 

where  r  >  0,  *  >  0,  p  ^  0,  /2(a)  9^  0. 

If  p  5^  0,  the  theorem  is  proved  since  X  —  a  is  a  factor  of  all  the  terms 
in  the  brackets  but  one  and  consequently  is  not  a  factor  of  the  sum.  Thus 
S'p  is  regular,  as  stated.  If  p  =  0,  we  must  show  that  mJ<i,(}C)  +  riiJ^QC)  is 
not  divisible  by  X  —  a  where  /2(a)  5^  0  and  /2(a)  9^  0,  i.e.,  we  must  show 
that  m  can  be  so  chosen  that  mf^ia)  +  fnf(a)  9^  0.  This  is  the  same 
condition  reached  before  and  is  satisfied  by  m  =  l/fiia).  Hence  we  may 
use  the  Jacobi  transformation  of  the  adjoint  form,  viz.,* 

,0,  lASij.         X'X'r'X".  .   J<»>z(«) 


i,j 


S    '  •       SS'        S'S"  5(n-i)5(») 


*  Muth,  I.e.,  pp.  70-72. 


258  ,'•','?•'''''♦•''    iJacSDoN  f '  Pairs  of  Hermitian  Forms . 

with  a  determinant  regular  with  respect  to  any  Hnear  factor,  and  where 

X'      =   SnUi  +  S21U2  +  8ziUz  +    •  •  •   +  8nlUn 
X"     =  S22U2  +  S32W3  +    •  •  •   +  Sn2Un 

(4)  r"  =  s'z'^m  +  •  •  •  +  s'n'^un 


(3)  To  decompose  the  general  term  on  the  right  of  (3)  with  respect  to 
the  real  linear  factor,  X  —  c,  we  may  write 

where  C^  is  the  coefficient  of  the  highest  power  of  X  in  S'^''~^^S^''\t  It  is 
evident  that  q^,  a  polynomial  in  X  with  real  coefficients,  will  not  contain 
X  —  c  as  a  factor  since  S^*~^^  and  S^"^  are  regular,  and  we  may  then  expand 
X'^'^^lq  and  X'^'^^Jq  in  power  series  in  X  —  c.  Also,  since  in  the  definition  of 
^w  in  (4)  i\^Q  coefficient  S^,""*^  of  u^  is  regular  with  respect  to  X  —  c,  Z^''^  and 
consequently  Z^*^  will  contain  X  —  c  exactly  l^  times.     Thus 

—  =  (X  -  c)'f  Xa  +  (X  -  c)X^2  +  (X  -  c)2Z.3  +•••], 

^  =  (X  -  c)'lZa  +  (X  -  c)Z.2  +  (X  -  cYX^z  +.••]• 
Hence 

^('<)X(«)         1        1  —  ■        — 

gc^D^w  =  C,(x-c)^'-^"^^'"^  (X-  c)Z,2H ][Za  +  (X  -  c)Z,2  +  •  •  -1 

where  the  Z^,^  are  linearly  independent  polynomials  in  c,  u^,  •  •  • ,  Un  and 
the  coefficients  of  the  two  forms  A  and  B  and  are  homogeneous  in  the  w's. 
Thus  dropping  the  subscript,  k,  we  may  write  the  right  member 

^(X^T^t^^i  +  Z.X\  -  c)  +  Z3(X  -  c)2  +  . . .] 

~  ^  C(X"^^2  -^  (7(x  _  c)  "^  "  '• 

Now,  defining  i'^,  =  Ze  =  Ll=iZ,Z,_,+i  and  G,  =  Z^-i  =  i:\=i-^XiX,-i 
{G^  =  0  for  e^  =  1)  we  have 


a(X-  c)      C,(X-c)2      c,Lx  X' 


*  For  convenience  in  notation  h  shall  henceforth  represent  the  exponent  of  the  factor 
X  —  c  In  the  greatest  common  divisor  of  the  (n  —  K)-rowed  minor  determinants  of  S. 
We  have  then  the  inequalities,  Zo  ^  ^1  =  ?2  =  •  •  •  =  h,,  with  U-i  —  /«  =  e*. 

t  See  Dickson,  I.e. 


Logsdon:   Pairs  of  Hermitiah'I^Ofm^X  *••  '*•*  :*•:  .•/•*2o9 

There  will  be  a  similar  expression  obtained  from  each  of  the  terms  on  the 
right  of  (3).  The  total  contribution  due  to  the  real  factor  X  —  c  to  the 
canonical  form  is  then  obtained  by  taking  the  sum 


where  /  is  the  number  of  distinct  real  linear  factors.  The  numbers  1/C, 
may  be  now  absorbed  in  the  variables  by  x'  =  xj-^C^  since  the  original 
forms  and  the  transformations  used  have  allowed  irrationalities  as  well  as 
imaginaries. 

If  now  the  above  process  of  regularization  and  expansion  be  repeated 
for  the  remaining  real  linear  factors,  the  total  contribution  of  these  factors 
is  obtained. 

To  decompose  with  respect  to  (X  —  a)(X  —  a)  the  general  term  of  (3), 
we  need  to  get  the  partial  fractions  of  this  term  which  have  linear  or 
quadratic  denominators,  viz.,  (X  —  a),  (X  —  a),  (X  —  of,  (X  —  o)^ 
(X  —  a)(X  —  a),  since  these  and  only  these  terms  in  the  decomposition  will 
contribute  to  the  coefficient  of  1/X  and  1/X^.     We  set 


S(^i)5W      ciQ,  -  a)  (X  -  a)]'«-i+'«g2  > 

where  q^  is  a  polynomial  in  X  containing  neither  X  —  a  nor  X  —  a  as  a  factor 
and  with  unity  for  the  coefficient  of  the  highest  power  of  X.  Now  expand 
X'^'^lq  in  powers  of  X  —  a  and  Z^^V?  in  powers  of  X  —  a.     Thus 

—  =  [(X  -  «)(X  -  a)]^l^a  +  (X  -  a)Z«2  +  (X  -  afX^z  +  •  •  •  J 

—  =  [(X  -  a)(X  -  a)]^f  Za  +  (X  -  a)Z.2  +  (X  -  ayX^z  +•••]. 

where  the  Z^^  are  polynomials  in  a,  a,  the  coefficients  of  A  and  B,  and 
homogeneous  in  w^,  •  •  • ,  w„,  while  the  Z,^  are  the  corresponding  conjugate 
functions  of  w»,  ••-,«„.     We  have  then 

W=^^^  =  C:(X-ar(X-arL^'"  +  ^^  "  ^^^'^  +  ^  "  ^^'^'"^ 

+  •  •  -I^.i  +  (X  -  a)X^,  +  (X  -  a)2Z,3  +•••]. 

Omitting  all  k  subscripts,  we  may  write  the  right  member 


1  r   Xi 

cL(X-a 


Z2         .  .      Z 


+  •••+,     '-z-\-  Xe+i  +  Z^2(X  -  a)  + 


Y      (X  -  d)*^!  X  -  d 


260/.'.:  ;';  /*I'.*  'i&OGsi'o^t*  Pairs  of  Hermitian  Forms. 

+  Z2e(X  -  a)-i  + 


;[(-x^+-+.-^+^-+ 


+  X2.(X-  a)^'  + 


Xe 

y  '        ■  X-  o 


1  rXiX2e(X  -  a)-i  ,  XiZ2e-i(X  -  tt)^'  .  X^X^e-iiX  "  a)"-^ 


—  n\o-l 


c\_  _  (K-ay      "^       (X -ay       "^      ix -  a) 

j^  A  2X26-2  (X  —   a)*~^j^                ,     XeXe+i     ,  XeXe 

■I    K ^T^Zl h   •  •  ■  -T  r ~  -T 


(X-d)"-^      _  X-a       (X-a)(X-o) 

,     Xe+lZe    ,  ,     X2eZi(X  —  O)^^     .  ~| 

-r  ;: i-  •  •  •  + 


X-a  (X-a)'  J 

~   PX  L-^l-^2e  +  ^'2X26-1  -|-    •  •  •    -j-  XeXe+1  +    '  •  '    +  XzeXiJ^ 

+  ii[a(ZiX2e  +    •  •  •    +  XeXe+l)  +  a(Ze+lZ«  +    •  •  •   +  X2eXt) 
~r   -3riA2e_l  +  X2X2e— 2  +    '  "  *    H"   X2e_iXiJ. 

Thus  we  have  found  the  part  of  the  coefficient  of  1/X  and  1/X^  due  to 
one  term  in  the  right  member  of  (3)  and  contributed  by  the  pair  of  linear 
factors  (X  —  a)(X  —  a)  where  the  e  in  the  last  expression  is  the  e^  which 
belongs  to  the  elementary  divisors  (X  —  a)%  (X  —  a)*".  The  total  coeffi- 
cient of  1/X  is  seen  to  be  (after  the  constant  C^  is  absorbed  in  the  variables) 

2ei  

22  XjX2e^-j+l 
J  =  l 

and  the  total  coefficient  of  1/X^  is 

«t       —  2e^         

2_^  aXjX2e^-j+l  +      2_/     ^XjX2e^-j+L- 
J=l  i=ek+l 

For  other  pairs  of  complex  elementary  divisors  we  proceed  as  here, 
then  adding  the  coefficients  of  1/X  obtained  from  all  the  linear  factors,  real 
and  imaginary,  and  adding  the  coefficients  of  1/X^  obtained  from  all  the 
Hnear  factors,  real  and  imaginary,  we  compare  with  the  coefficients  of  these 
same  powers  of  X  obtained  by  expanding  the  adjoint  form 

^■■^  Kj  -ij  Uj  (4/% 

by  determinanta    methods,  *  thus  obtaining  the  desired  canonical  forms 

given  in  Part  I,  Theorem  V. 

The  University  of  Chicago, 
May,  1921. 

*  Muth,  I.e.,  p.  81.       2       g      ^x'^x^^'"- 


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